Mathematics in Ancient and Medieval India c5.s8: Eight Fundamental Operations

Mathematics in Ancient and Medieval India c5.s8

Language: English

Eight Fundamental Operations

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The eight fundamental operations of Indian arithmetic after the invention of the decimal place value system of numeration are : addition (*samkalita*), subtraction ( *vyavakalita* or *vyutkalita* ), multiplication (*guṇana*), division (*bhāgahāra*), square (*varga*), square-root ( *varga-mūla* ), cube ( *ghana* ), cube-root ( *ghana-mūla*).

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These operations were carried out on a dust computing board (*pāṭī*). The method required the rubbing out of digits in every operation. According to Arabian scholars Āl-Khwārizmī (825 A. D.), Al-Uqlidisi (952 A. D.) Hindu arithmetic entered Islam with dust abacus as an intrinsic tool of it.² The latter author replaced the dust abacus to suit paper and ink. He used the Arabic word *takht* for *pāṭī*.

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² Datta, B. 'Testimony of early Arab writers on the origin of our numerals', *Bulletin of the Calcutta Mathematical Society*, 24, p. 193-218, 1932; Saidan, A. S. 'The development of Hindu Arabic Arithmetic', *Islamic Culture*, 39, p. 209-221, 1965.

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The ancient Indian mathematical works have not devoted much space to the methods of addition and subtraction in view of simple and self evident character, though such methods were described in details in the *Līlāvatī* of Bhāskara II ( 1150 A. D. ) and in two commentaries of *Līlāvatī* namely *Gaṇitāmṛta* of Sūryadāsa (1538 A. D.) and *Manorañjanā* of Rāmakṛṣṇa. Several methods of multiplication namely *kapāṭasandhi*, *gomūtrikā*, *khaṇḍa*, *bheda* and *iṣṭa* etc. were known. Datta and Singh have described this method fully in their *History of Hindu Mathematics*¹ as it appeared in the works of Mahāvīra (c. 850 A.D.), Śrīdhara (900 A.D.) and Śrīpati (1039 A.D.). The main feature of the method of *kapāṭasandhi* was that it was carried out in two stages : (1) the relative positions of the multiplicand and the multiplier followed by, (2) the rubbing out of figures of the multiplicand and substitution of the figures of the product in their places. The *kapāṭa* and *sandhi* owe their origin from the first and second stages respectively. The terms *hanana*, *vadha* or *kṣaya* which mean killing' or ‘destroying’ have been used in the works of Brahmagupta² (628 A. D.) in describing his process of multiplication. It seems to suggest that this method of *kapāṭasandhi* was known to India in 7th century A.D. This method reappeared in works of the Arabian scholars³ Al-Nasabī (c. 1025 A. D.), Al-Ḥaṣṣār (c. 1175 A. D.), Al-Kalasādi (c. 1475 A. D.).

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¹ *I*, p. 134-149, 1935.

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² Brahmagupta mentions four methods of multiplication viz. *gomūtrikā*, *khaṇḍa*, *bheda* and *iṣṭa* (vide Datta and Singh, *Ibid.*, *1*, p. 135).

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³ Datta and Singh, *Ibid.*, *1*, p. 143.

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The method of division in India was given by Mahāvīra (850 A. D.), Śrīdhara (900 A. D.), Āryabhaṭa II (950 A. D.), Bhāskara II (1150 A. D.), Nārāyaṇa (1356 A. D.) and others.⁴ The partial divisions were done and partial quotients were written in a separate line on the dust board and the figures of these operations were obliterated. If the figures were not obliterated and the successive steps were written one below the other, this ancient process of division would become the modern method of long division. Āryabhaṭa I (499 A. D.) gave the method of extracting square root and cube root which depend on the method of division. This seems to suggest that the method originated in India about the 4th century A. D. It reappeared in the Arabian works¹ from 9th century onwards. From Arabia, the method travelled to Europe where it came to be known as the galley (*galea*, *batello*) method. Fundamental operations for square and cube as developed by ancient and medieval Indian workers have been discussed fully by Datta and Singh.²

Linked texts (1)

Līlāvaī — Kouhala, Māhārāṣṭrī Prakrit